We investigate the Markov property and the continuity with respect to the initial conditions (strong Feller property) for the solutions to the Navier-Stokes equations forced by an additive noise. First, we prove, by means of an abstract selection principle, that there are Markov solutions to the Navier-Stokes equations. Due to the lack of continuity of solutions in the space of finite energy, the Markov property holds almost everywhere in time. Then, depending on the regularity of the noise, we prove that any Markov solution has the strong Feller property for regular initial conditions. We give also a few consequences of these facts, together with a new sufficient condition for well-posedness.

We prove global existence of smooth solutions for a slightly supercritical hyperdissipative Navier-Stokes under the optimal condition on the correction to the dissipation. This proves a conjecture formulated by Tao [Tao09]. From the generalized Fourier Navier-Stokes to the dyadic equationThis section contains one of the crucial steps in our approach. We show that the proof of Theorem 1.2 can be reduced to a proof of decay of solutions of a suitable

We prove ergodicity of the finite dimensional approximations of the three dimensional Navier-Stokes equations, driven by a random force. The forcing noise acts only on a few modes and some algebraic conditions on the forced modes are found that imply the ergodicity. The convergence rate to the unique invariant measure is shown to be exponential.2000 Mathematics Subject Classification. Primary 76D05; Secondary 35Q30, 76M35, 76F55. 1 2 M. ROMITOBoth these properties, strong Feller and irreducibility, are implied by an algebraic condition on the set of indices corresponding to the modes forced by the noise. The condition essentially means that it is possible to obtain any index as a sum of some of the forced indices. One can see this mechanism as a geometrical realisation of the cascade of energy, since the non-linear term transmits the random forcing from the few forced modes to all the other modes. As an example we show that the algebraic condition is satisfied if the three lowest modes are forced.Recently, many authors have applied the techniques we have used here, such us the hypoellipticity for degenerate diffusions, or the general theory for Markov chains and Markov processes collected and developed by Meyn and Tweedie (one can see their book [10]). Among many others we quote the papers by E and Mattingly [2], Eckmann and Hairer [3], Hairer [6], Rey-Bellet and Thomas [13], and some of the references therein.The paper is organised as follows. In the first section the main definitions are given, together with the statements of the main results and an outline of their proofs. The technical computations and the precise statement of some hypotheses are postponed in the following sections. The aim is to give a light presentation of the main ideas, without all the technicalities, which are then reserved to the interested readers.Acknowledgements. The author wish to thanks R. Bianchini and S. Dolfi for the helpful bibliographical suggestions on the control theory part in Section 6 and on the algebraic part in Section 5, and F. Flandoli for the many helpful conversations. This paper is dedicated to the memory of my father, who died whilst I was writing it. The main theoremWe consider the stochastic Navier-Stokes equations with additive noise du = (ν∆u − (u · ∇)u − ∇P ) dt + dB t div u = 0,

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We prove three results on the existence of densities for the laws of finite dimensional functionals of the solutions of the stochastic Navier-Stokes equations in dimension 3. In particular, under very mild assumptions on the noise, we prove that finite dimensional projections of the solutions have densities with respect to the Lebesgue measure which have some smoothness when measured in a Besov space. This is proved thanks to a new argument inspired by an idea introduced in [18]